Finite Group Databasebeta

← #28345959

Subgroups — #28345959

Order 13696.

526 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812133C211yesmaximal, characteristic, complemented
2C2×D8×D107C211yesmaximal, characteristic, complemented
3#20293190C211yesmaximal, characteristic, complemented
4#20293099C211yesmaximal, characteristic, complemented
5#20292843C211yesmaximal, characteristic, complemented
6#20292842C211yesmaximal, characteristic, complemented
7#20292841C211yesmaximal, characteristic, complemented
8#20292840C211yesmaximal, characteristic, complemented
9#20292821C211yesmaximal, characteristic, complemented
10#20292820C211yesmaximal, characteristic, complemented
11#20292819C211yesmaximal, characteristic, complemented
12#20292652C211yesmaximal, characteristic, complemented
13#20292630C211yesmaximal, characteristic, complemented
14#20292620C211yesmaximal, characteristic, complemented
15#459×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×D4×D107C2211yescharacteristic
18C2×D4×D107C2211yescharacteristic
19#16812133—24no—
20C2×#13133807C2211yescharacteristic, complemented
21C2×#13133806C2211yescharacteristic, complemented
22C2×D856C2211yescharacteristic
23C2×#13133803C2211yescharacteristic
24C2×#13133803C2211yescharacteristic
25C2×#13133803C2211yescharacteristic, complemented
26D8×D107—24no—
27C2×#13133801C2211yescharacteristic
28C2×C8×D107C2211yescharacteristic
29#16812105C2211yescharacteristic, complemented
30#16812097C2211yescharacteristic, complemented
31#16812096C2211yescharacteristic, complemented
32#16812081—21no—
33#16812081—21no—
34#16812077—21no—
35#16812077—21no—
36C4×D428C2211yescharacteristic, complemented
37D4×Dic107C2211yescharacteristic, complemented
38#16812071C2211yescharacteristic, complemented
39C4×C22⋊D107—21no—
40(C22⋊D107)⋊C4—21no—
41#16812067C2211yescharacteristic
42#16812066C2211yescharacteristic
43C2×(C2×D107)⋊C4—21no—
44C22⋊C4×D107—21no—
45C4⋊(C107⋊C8)C2211yescharacteristic, complemented
46Dic107⋊C8C2211yescharacteristic, complemented
47Dic107⋊C8C2211yescharacteristic, complemented
48#16812034C2211yescharacteristic, complemented
49#16812032C2211yescharacteristic, complemented
50#16812031C2211yescharacteristic, complemented
51#16812030C2211yescharacteristic, complemented
52#16812030C2211yescharacteristic, complemented
53#16812029C2211yescharacteristic, complemented
54C2×D8×C107C2211yescharacteristic
55C2×D8×C107C2211yescharacteristic, complemented
56#127×C107C2211yescharacteristic, complemented
57C4×D4×C107C2211yescharacteristic, complemented
58C4⋊C8×C107C2211yescharacteristic, complemented
59#102×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C23×D107—21no—
63#13133817—22no—
64#13133817—22no—
65C2×Dic214C2311yescharacteristic
66C2×D428C2311yescharacteristic
67C2×D428C2311yescharacteristic
68C2×C22⋊D107—21no—
69C2×C22⋊D107—21no—
70C2×C22⋊D107—21no—
71C2×C22⋊D107—21no—
72C2×C22⋊D107—21no—
73D4×D107—22no—
74D4×D107—22no—
75D4×D107—22no—
76D4×D107D414yescomplemented
77C22×Dic107—21no—
78C22×Dic107—21no—
79C2×C4×D107—21no—
80C2×C4×D107—21no—
81C2×C4×D107C2311yescharacteristic
82#13133807—22no—
83#13133806—22no—
84D856—22no—
85#13133803—22no—
86#13133803—22no—
87#13133803—22no—
88#13133801—22no—
89C2×C107⋊C8C2311yescharacteristic
90C2×C107⋊C8C2311yescharacteristic
91C8×D107—22no—
92C4⋊Dic107C2311yescharacteristic
93Dic107⋊C4D411yescharacteristic
94Dic107⋊C4D411yescharacteristic
95C22⋊Dic107—21no—
96C22⋊Dic107—21no—
97C22⋊Dic107—21no—
98(C2×D107)⋊C4—21no—
99(C2×D107)⋊C4—21no—
100(C2×D107)⋊C4—21no—
101(C2×D107)⋊C4—21no—
102(C2×D107)⋊C4—21no—
103(C2×D107)⋊C4D411yescharacteristic
104(C2×D107)⋊C4D411yescharacteristic
105C4×Dic107—21no—
106C4×Dic107C2311yescharacteristic
107C2×D4×C107—21no—
108C2×D4×C107—21no—
109C2×D4×C107C2311yescharacteristic
110C2×D4×C107C2311yescharacteristic
111C2×D4×C107C2311yescharacteristic
112C22×C4×C107—21no—
113D8×C107—22no—
114D8×C107—22no—
115C2×C8×C107C2311yescharacteristic
116C2×C8×C107C2311yescharacteristic
117C4⋊C4×C107C2311yescharacteristic
118C22⋊C4×C107—21no—
119C42×C107C2311yescharacteristic
120C23×C107—21no—
121C23×C107—21no—
122C23×C107—21no—
123C22×D107—21no—
124C22×D107—24no—
125C22×D107—21no—
126C22×D107—42no—
127C22×D107—81no—
128C22×D107—81no—
129C22×D107—81no—
130C22×D107—81no—
131C22×D107C2×D411yescharacteristic
132D4×C107—21no—
133D4×C107—21no—
134D4×C107—21no—
135D4×C107—21no—
136D4×C107—21no—
137D4×C107—81no—
138D4×C107—81no—
139D4×C107C2×D412yes—
140C2×C4×C107—21no—
141C2×C4×C107—81no—
142C2×C4×C107C2×D411yescharacteristic
143C2×C4×C107C2×D411yescharacteristic
144C2×C4×C107C2411yescharacteristic
145C22⋊D107—24no—
146C22⋊D107—42no—
147C22⋊D107—42no—
148C22⋊D107—81no—
149C22⋊D107—81no—
150C2×Dic107—21no—
151C2×Dic107—21no—
152C2×Dic107—42no—
153C2×Dic107—81no—
154C2×Dic107C2×D411yescharacteristic
155C2×Dic107C2×D411yescharacteristic
156C2×Dic107C2×D411yescharacteristic
157D428—21no—
158D428—21no—
159D428—21no—
160D428C2×D412yes—
161C4×D107—81no—
162C4×D107—81no—
163C4×D107C2×D412yes—
164C4×D107#4112yes—
165Dic214—21no—
166Dic214—21no—
167C8×C107—21no—
168C8×C107—21no—
169C107⋊C8—21no—
170C107⋊C8—21no—
171C22×C107—22no—
172C22×C107—41no—
173C22×C107—41no—
174C22×C107—41no—
175C22×C107—41no—
176C22×C107—41no—
177C22×C107—41no—
178C22×C107—41no—
179C22×C107—41no—
180C22×C107C22×D411yescharacteristic
181C2×D107—21no—
182C2×D107—21no—
183C2×D107—22no—
184C2×D107—41no—
185C2×D107—41no—
186C2×D107—41no—
187C2×D107—41no—
188C2×D107—41no—
189C2×D107—84no—
190C2×D107—84no—
191C2×D107#12112yes—
192C4×C107—21no—
193C4×C107—21no—
194C4×C107—41no—
195C4×C107C22×D411yescharacteristic
196C4×C107C2×#4111yescharacteristic
197Dic107—21no—
198Dic107—21no—
199Dic107—41no—
200Dic107—41no—
201Dic107#12112yes—
202C2×C107—42no—
203C2×C107—81no—
204C2×C107—81no—
205C2×C107C2×#12111yescharacteristic
206C2×C107#57611yescharacteristic
207C2×C107#57611yescharacteristic
208D107—22no—
209D107—42no—
210D107—81no—
211#3041—1071nomaximal, Sylow
212C107#304111yescharacteristic, complemented, Sylow
213C2×AGL(1,ℤ/8)—1071no—
214C22×D8—1071no—
215#535—1071no—
216#518—1071no—
217#463—1071no—
218#463—1071no—
219#463—1071no—
220#463—1071no—
221#459—1071no—
222#459—1071no—
223#459—1071no—
224#459D10711yescharacteristic, complemented
225#424—1071no—
226#418—1071no—
227#416—1071no—
228C2×#41—1071no—
229C22×D4—1071no—
230C22×D4—1071no—
231AGL(1,ℤ/8)—2144no—
232C2×SD16—1071no—
233C2×SD16—1071no—
234C2×D8—1071no—
235C2×D8—1071no—
236C2×D8—1071no—
237C2×D8—1071no—
238C2×D8—2144no—
239C2×D8C2×D10711yescharacteristic
240C2×D8C2×D10711yescharacteristic, complemented
241C2×M16—1071no—
242C22×C8—1071no—
243#127—1071no—
244#127C2×D10711yescharacteristic, complemented
245#124—1071no—
246#124—1071no—
247#121—2141no—
248#121—2141no—
249#120—2141no—
250#120—2141no—
251C4×D4—1071no—
252C4×D4—1071no—
253C4×D4—1071no—
254C4×D4—2141no—
255C4×D4—2141no—
256C4×D4C2×D10711yescharacteristic, complemented
257#117—1071no—
258#117—1071no—
259C2×C22⋊C4—2141no—
260C2×C22⋊C4—2141no—
261C4⋊C8—1071no—
262C4⋊C8—1071no—
263C4⋊C8—1071no—
264C4⋊C8C2×D10711yescharacteristic, complemented
265#103—1071no—
266#103—1071no—
267#102—1071no—
268#102—1071no—
269#102—1071no—
270#102—1071no—
271#102C2×D10711yescharacteristic, complemented
272#102C2×D10711yescharacteristic, complemented
273C24—2141no—
274C24—2141no—
275#41—2142no—
276#41—2142no—
277C2×Q8—1071no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—1074no—
281C2×D4—1071no—
282C2×D4—1071no—
283C2×D4—2142no—
284C2×D4—2142no—
285C2×D4—2142no—
286C2×D4—2141no—
287C2×D4—2141no—
288C2×D4—2141no—
289C2×D4—2141no—
290C2×D4—2141no—
291C2×D4C22×D10711yescharacteristic
292C2×D4C22×D10711yescharacteristic
293C2×D4C22×D10711yescharacteristic
294C22×C4—21no—
295C22×C4—1071no—
296C22×C4—2141no—
297C22×C4—2141no—
298C22×C4—2141no—
299C22×C4—2141no—
300SD16—2142no—
301SD16—2142no—
302D8—22no—
303D8—22no—
304D8—2142no—
305D8—2142no—
306D8—2142no—
307D8—2142no—
308M16—2142no—
309C2×C8—1071no—
310C2×C8—1071no—
311C2×C8—2142no—
312C2×C8C22×D10711yescharacteristic
313C2×C8C22×D10711yescharacteristic
314C4⋊C4—1071no—
315C4⋊C4—1071no—
316C4⋊C4—1071no—
317C4⋊C4C22×D10711yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—1071no—
320C22⋊C4—1071no—
321C22⋊C4—2141no—
322C22⋊C4—2141no—
323C22⋊C4—2141no—
324C22⋊C4—2141no—
325C22⋊C4—2141no—
326C22⋊C4—2141no—
327C22⋊C4—2141no—
328C22⋊C4—2141no—
329C42—1071no—
330C42—2141no—
331C42C22×D10711yescharacteristic
332C23—21no—
333C23—21no—
334C23—21no—
335C23—8561no—
336C23—8561no—
337C23—8561no—
338C23—8561no—
339C23—1071no—
340C23—4282no—
341C23—2144no—
342C23—2141no—
343C23—2141no—
344Q8—2141no—
345Q8—2141no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—81no—
352D4—81no—
353D4—8561no—
354D4—8561no—
355D4—1072no—
356D4—4282no—
357D4—4282no—
358D4—2144no—
359D4—2141no—
360D4—2141no—
361D4—2141no—
362D4D4×D10712yes—
363C2×C4—21no—
364C2×C4—81no—
365C2×C4—8561no—
366C2×C4—8561no—
367C2×C4—8561no—
368C2×C4—1071no—
369C2×C4—1072no—
370C2×C4—1072no—
371C2×C4—1071no—
372C2×C4—1071no—
373C2×C4—4282no—
374C2×C4—2141no—
375C2×C4—2141no—
376C2×C4D4×D10711yescharacteristic
377C2×C4D4×D10711yescharacteristic
378C2×C4C23×D10711yescharacteristic
379C8—21no—
380C8—21no—
381C8—2141no—
382C8—2141no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—8564no—
393C22—8564no—
394C22—1072no—
395C22—4281no—
396C22—4281no—
397C22—4281no—
398C22—4281no—
399C22—4281no—
400C22—2142no—
401C22—2141no—
402C22—2141no—
403C22C2×D4×D10711yescharacteristic
404C4—21no—
405C4—21no—
406C4—41no—
407C4—1072no—
408C4—4281no—
409C4—4281no—
410C4—2141no—
411C4—2141no—
412C4C2×D4×D10711yescharacteristic
413C4#41×D10711yescharacteristic
414C2—42no—
415C2—81no—
416C2—81no—
417C2—8561no—
418C2—4282no—
419C2—2142no—
420C2#121×D10711yescharacteristic
421C2#2029342811yescharacteristic
422C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#459×C107C2
socleC22×C107C22×D4
hypercenter#459D107
hypocenterC107#3041
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16812133C2semidirect
2. C2×D8×D107C2semidirect
3. #20293190C2semidirect
4. #20293099C2semidirect
5. #20292843C2semidirect
6. #20292842C2semidirect
7. #20292841C2semidirect
8. #20292840C2semidirect
9. #20292821C2semidirect
10. #20292820C2semidirect
11. #20292819C2semidirect
12. #20292652C2semidirect
13. #20292630C2semidirect
14. #20292620C2semidirect
15. #459×C107C2semidirect
20. C2×#13133807C22semidirect
21. C2×#13133806C22semidirect
25. C2×#13133803C22semidirect
29. #16812105C22semidirect
30. #16812097C22semidirect
31. #16812096C22semidirect
36. C4×D428C22semidirect
37. D4×Dic107C22semidirect
38. #16812071C22semidirect
45. C4⋊(C107⋊C8)C22semidirect
46. Dic107⋊C8C22semidirect
47. Dic107⋊C8C22semidirect
48. #16812034C22semidirect
49. #16812032C22semidirect
50. #16812031C22semidirect
51. #16812030C22semidirect
52. #16812030C22semidirect
53. #16812029C22semidirect
55. C2×D8×C107C22semidirect
56. #127×C107C22semidirect
57. C4×D4×C107C22semidirect
58. C4⋊C8×C107C22semidirect
59. #102×C107C22semidirect
60. #102×C107C22semidirect
76. D4×D107D4semidirect
212. C107#3041semidirect
224. #459D107semidirect
240. C2×D8C2×D107semidirect
244. #127C2×D107semidirect
256. C4×D4C2×D107semidirect
264. C4⋊C8C2×D107semidirect
271. #102C2×D107semidirect
272. #102C2×D107semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168121332
C2×D8×D1072
#202931902
#202930992
#202928432
#202928422
#202928412
#202928402
#202928212
#202928202
#202928192
#202926522
#202926302
#202926202
#459×C1072
#3041107